In orbital mechanics and aerospace engineering a gravitational slingshot (also known as gravity assist manoeuver or swing-by) is the use of the relative movement and gravity of a planet or other celestial body to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense. Gravity assistance can be used to accelerate (both positively and negatively) and/or re-direct the path of spacecraft.

Over-simplified example of gravitational slingshot: the spacecraft's velocity changes by up to twice the planet's velocity.

2 Answers
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If you analyse the situation from the planetary frame of reference (which will essentially be the centre-of-mass frame), then the incoming probe has a velocity $U+v$ if it then does any sort of "elastic bounce" off the planet - i.e. if it loses no energy and goes back in the opposite direction - then the planet will not be affected (if it's massive enough) and the probe will go back in the opposite direction and with the same speed, $U+v$, as initially. If you add to this the planet's velocity $U$, then you get a speed of $2U+v$ in the original frame of reference.

The emergence of the factor of 2 is easy to be seen from the celestial body's vantage point (in its reference frame).

Before the maneuver, the spaceship is moving by the velocity $\vec v-\vec u$ – the relative speed of the two objects – in this frame. The celestial body is largely unaffected so after the slingshot maneuver, we are still in the same frame.

Clearly, by the $Z_2$ symmetry of the situation, the final velocity of the spaceship is just minus the initial one, $\vec u-\vec v$. This velocity may be translated to the original reference frame by adding $\vec u$ that we subtracted at the beginning to get $\vec v -\vec u$ from $\vec v$. So we get $2\vec u-\vec v$ as the final velocity in the initial frame.

The signs are so that $\vec u$ has a negative $x$ component so both terms in $2\vec u-\vec v$ have a negative $x$ component and the picture represents the absolute value of the final $v_x$ which is therefore $2|u_x|+|v_x|$.